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<h1>Grade 9 Math Exam Study Guide</h1>
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<ul>
<li><a href="#ratios">Ratios</a></li>
<li><a href="#percentages">Percentages</a></li>
<li><a href="#notation">Scientific Notation</a></li>
<li><a href="#cartesian">Cartesian Plane</a></li>
<li><a href="#squares">Sum of Squares</a></li>
<li><a href="#graphing">Graph a Line</a></li>
<li><a href="#slope">Slope & Intercept</a></li>
<li><a href="#linear-system">Linear Systems</a></li>
<li><a href="#scatter-plot">Scatter Plot</a></li>
<li><a href="#short-answer">Short Answer</a></li>
<li><a href="#choice">Choice</a></li>
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<section id="main">
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<article id="ratios">
<h2>Ratios</h2>
<p><strong>Key Points:</strong></p>
<ul>
<li>A ratio compares two quantities.</li>
<li>Simplify ratios by dividing both terms by their greatest common divisor.</li>
</ul>
<p><strong>Example:</strong></p>
<p>Simplify the ratio 18:24.</p>
<p>\[ \frac{18}{24} = \frac{3}{4} \]</p>
<p>So, the simplified ratio is 3:4.</p>
<p><strong>Practice:</strong></p>
<ul>
<li>Simplify the ratio 45:60.</li>
<li>If a recipe calls for 3 cups of flour to 2 cups of sugar, write the ratio of flour to sugar.</li>
</ul>
</article>
<article id="percentages">
<h2>Percentages</h2>
<p><strong>Key Points:</strong></p>
<ul>
<li>Percentage means "per hundred."</li>
<li>Convert between fractions, decimals, and percentages.</li>
<li>Calculate percentages of a number.</li>
</ul>
<p><strong>Example:</strong></p>
<p>What is 20% of 150?</p>
<p>\[ 20\% \times 150 = 0.20 \times 150 = 30 \]</p>
<p><strong>Practice:</strong></p>
<ul>
<li>Convert 3/4 to a percentage.</li>
<li>What is 15% of 250?</li>
</ul>
</article>
<article id="notation">
<h2>Scientific Notation</h2>
<p><strong>Key Points:</strong></p>
<ul>
<li>Scientific notation expresses numbers as a product of a number between 1 and 10 and a power of 10.</li>
<li>Standard notation is the usual way of writing numbers.</li>
</ul>
<p><strong>Example:</strong></p>
<p>Convert 4500 to scientific notation.</p>
<p>\[ 4500 = 4.5 \times 10^3 \]</p>
<p><strong>Practice:</strong></p>
<ul>
<li>Convert 0.00034 to scientific notation.</li>
<li>Convert \( 3.6 \times 10^4 \) to standard notation.</li>
</ul>
</article>
<article id="cartesian">
<h2>Plot Points on Cartesian Plane</h2>
<p><strong>Key Points:</strong></p>
<ul>
<li>Points are written as (x, y).</li>
<li>The Cartesian plane has four quadrants.</li>
</ul>
<p><strong>Example:</strong></p>
<p>Plot the point (3, -2).</p>
<p><strong>Practice:</strong></p>
<ul>
<li>Plot the point (-4, 5).</li>
<li>Identify the coordinates of a point in Quadrant II.</li>
</ul>
</article>
<article id="squares">
<h2>Sum of Squares</h2>
<p><strong>Key Points:</strong></p>
<ul>
<li>Sum of squares formula for first n natural numbers: \( \sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6} \).</li>
</ul>
<p><strong>Example:</strong></p>
<p>Find the sum of squares of the first 3 natural numbers.</p>
<p>\[ 1^2 + 2^2 + 3^2 = 1 + 4 + 9 = 14 \]</p>
<p><strong>Practice:</strong></p>
<ul>
<li>Find the sum of squares of the first 5 natural numbers.</li>
<li>Find the sum of squares of the numbers 4, 5, and 6.</li>
</ul>
</article>
<article id="graphing">
<h2>Graph a Line Given Equation</h2>
<p><strong>Key Points:</strong></p>
<ul>
<li>The equation of a line in slope-intercept form is \( y = mx + b \), where m is the slope and b is the y-intercept.</li>
</ul>
<p><strong>Example:</strong></p>
<p>Graph the line \( y = 2x + 3 \).</p>
<p><strong>Practice:</strong></p>
<ul>
<li>Graph the line \( y = -x + 2 \).</li>
<li>Graph the line \( y = \frac{1}{2}x - 4 \).</li>
</ul>
</article>
<article id="slope">
<h2>Determine Slope and Y-intercept from an Equation</h2>
<p><strong>Key Points:</strong></p>
<ul>
<li>In the equation \( y = mx + b \), m is the slope and b is the y-intercept.</li>
</ul>
<p><strong>Example:</strong></p>
<p>For \( y = -3x + 5 \), the slope is -3 and the y-intercept is 5.</p>
<p><strong>Practice:</strong></p>
<ul>
<li>Determine the slope and y-intercept of \( y = 4x - 7 \).</li>
<li>Determine the slope and y-intercept of \( y = \frac{3}{2}x + 1 \).</li>
</ul>
</article>
<article id="linear-system">
<h2>Determine Slope and Y-intercept from a Graph</h2>
<p><strong>Key Points:</strong></p>
<ul>
<li>Slope is the rise over run.</li>
<li>The y-intercept is where the line crosses the y-axis.</li>
</ul>
<p><strong>Example:</strong></p>
<p>If a line passes through (0, 2) and (3, 8), the slope is \( \frac{8-2}{3-0} = 2 \) and the y-intercept is 2.</p>
<p><strong>Practice:</strong></p>
<ul>
<li>Determine the slope and y-intercept of a line passing through (2, 5) and (4, 9).</li>
<li>Determine the slope and y-intercept of a line passing through (0, -3) and (2, 1).</li>
</ul>
</article>
<article id="scatter-plot">
<h2>State the Solution to a Linear System from a Graph</h2>
<p><strong>Key Points:</strong></p>
<ul>
<li>The solution is the point where the two lines intersect.</li>
</ul>
<p><strong>Example:</strong></p>
<p>If lines \( y = 2x + 1 \) and \( y = -x + 4 \) intersect at (1, 3), then (1, 3) is the solution.</p>
<p><strong>Practice:</strong></p>
<ul>
<li>Determine the solution from the graph of \( y = x + 2 \) and \( y = -2x + 5 \).</li>
<li>Find the intersection of \( y = \frac{1}{2}x - 1 \) and \( y = -x + 4 \).</li>
</ul>
</article>
<article id="scatter-plot">
<h2>Scatter Plot, Independent/Dependent Variable, Trend, Strength</h2>
<p><strong>Key Points:</strong></p>
<ul>
<li>Independent variable: horizontal axis.</li>
<li>Dependent variable: vertical axis.</li>
<li>Trend: general direction (positive, negative).</li>
<li>Strength: how closely data points fit the trend line.</li>
</ul>
<p><strong>Example:</strong></p>
<p>Plot and analyze the scatter plot of data points: (1, 2), (2, 4), (3, 6), (4, 8).</p>
<p><strong>Practice:</strong></p>
<ul>
<li>Create a scatter plot for the points: (2, 3), (3, 5), (5, 10), (6, 12).</li>
<li>Determine the trend and strength of the data.</li>
</ul>
</article>
<article id="short-answer">
<h2>Short Answer – 17 marks</h2>
<h3>Identify Linear/Non-linear Relation from Equation</h3>
<p><strong>Key Points:</strong></p>
<ul>
<li>Linear: equations of the form \( y = mx + b \).</li>
<li>Non-linear: equations involving squares, cubes, etc. (e.g., \( y = x^2 + 3 \)).</li>
</ul>
<p><strong>Example:</strong></p>
<p>Identify if \( y = 3x^2 + 2x \) is linear or non-linear.</p>
<p>Non-linear, because it includes \( x^2 \).</p>
<p><strong>Practice:</strong></p>
<ul>
<li>Is \( y = 4x - 7 \) linear or non-linear?</li>
<li>Is \( y = x^3 - 2x + 1 \) linear or non-linear?</li>
</ul>
<h3>Determine Linear Relation from Table of Values / Fill in Missing Values</h3>
<p><strong>Key Points:</strong></p>
<ul>
<li>Linear relations have constant differences in y-values for equal changes in x-values.</li>
</ul>
<p><strong>Example:</strong></p>
<p>Determine if the table represents a linear relation:</p>
<table>
<tr><th>x</th><th>y</th></tr>
<tr><td>1</td><td>3</td></tr>
<tr><td>2</td><td>5</td></tr>
<tr><td>3</td><td>7</td></tr>
<tr><td>4</td><td>9</td></tr>
</table>
<p>Yes, because the change in y is constant (2).</p>
<p><strong>Practice:</strong></p>
<ul>
<li>Fill in missing value for the linear relation:</li>
</ul>
<table>
<tr><th>x</th><th>y</th></tr>
<tr><td>1</td><td>2</td></tr>
<tr><td>2</td><td>4</td></tr>
<tr><td>3</td><td>?</td></tr>
<tr><td>4</td><td>8</td></tr>
</table>
<ul>
<li>Is this a linear relation?</li>
</ul>
<table>
<tr><th>x</th><th>y</th></tr>
<tr><td>1</td><td>1</td></tr>
<tr><td>2</td><td>4</td></tr>
<tr><td>3</td><td>9</td></tr>
<tr><td>4</td><td>16</td></tr>
</table>
<h3>Graph Linear Inequality</h3>
<p><strong>Key Points:</strong></p>
<ul>
<li>Similar to graphing a line but shade the appropriate side of the boundary line.</li>
<li>Use dashed lines for < or > and solid lines for ≤ or ≥.</li>
</ul>
<p><strong>Example:</strong></p>
<p>Graph \( y < 2x + 3 \).</p>
<p><strong>Practice:</strong></p>
<ul>
<li>Graph \( y \geq -x + 1 \).</li>
<li>Graph \( y \leq \frac{1}{2}x - 2 \).</li>
</ul>
<h3>Perimeter/Area of 2D Composite Shape</h3>
<p><strong>Key Points:</strong></p>
<ul>
<li>Break the shape into simpler shapes.</li>
<li>Use formulas for perimeter and area of rectangles, triangles, etc.</li>
</ul>
<p><strong>Example:</strong></p>
<p>Find the area of a shape composed of a rectangle (3x4) and a triangle (base 4, height 3).</p>
<p><strong>Practice:</strong></p>
<ul>
<li>Find the perimeter of a shape made of two rectangles (5x2 and 3x2).</li>
<li>Find the area of a shape made of a rectangle (6x4) and a semicircle (radius 2).</li>
</ul>
<h3>Volume of 3D Shape</h3>
<p><strong>Key Points:</strong></p>
<ul>
<li>Use formulas for volume of common shapes (e.g., cube \( V = a^3 \), cylinder \( V = \pi r^2 h \)).</li>
</ul>
<p><strong>Example:</strong></p>
<p>Find the volume of a cylinder with radius 3 and height 5.</p>
<p>\[ V = \pi (3)^2 (5) = 45\pi \]</p>
<p><strong>Practice:</strong></p>
<ul>
<li>Find the volume of a cube with side length 4.</li>
<li>Find the volume of a rectangular prism with dimensions 2x3x4.</li>
</ul>
<h3>Determine Missing Angle Values, Including Theorem Used</h3>
<p><strong>Key Points:</strong></p>
<ul>
<li>Use angle sum properties (e.g., sum of angles in a triangle is 180°).</li>
<li>Use theorems like corresponding, alternate interior angles, etc.</li>
</ul>
<p><strong>Example:</strong></p>
<p>Find the missing angle in a triangle with angles 50° and 60°.</p>
<p>\[ 50 + 60 + x = 180 \Rightarrow x = 70° \]</p>
<p><strong>Practice:</strong></p>
<ul>
<li>Find the missing angle in a triangle with angles 45° and 85°.</li>
<li>Find the missing angle in a quadrilateral with angles 90°, 80°, and 95°.</li>
</ul>
</article>
<article id="choice">
<h2>Choice - 18 marks</h2>
<h3>Simplify an Expression Using Exponent Laws</h3>
<p><strong>Key Points:</strong></p>
<ul>
<li>Use laws: \( a^m \cdot a^n = a^{m+n} \), \( \frac{a^m}{a^n} = a^{m-n} \), \( (a^m)^n = a^{mn} \).</li>
</ul>
<p><strong>Example:</strong></p>
<p>Simplify \( 2^3 \cdot 2^4 \).</p>
<p>\[ 2^3 \cdot 2^4 = 2^{3+4} = 2^7 \]</p>
<p><strong>Practice:</strong></p>
<ul>
<li>Simplify \( x^2 \cdot x^3 \).</li>
<li>Simplify \( \frac{y^5}{y^2} \).</li>
</ul>
<h3>Simplify a Polynomial (Add/Subtract/Distributive Property)</h3>
<p><strong>Key Points:</strong></p>
<ul>
<li>Combine like terms.</li>
<li>Use distributive property \( a(b + c) = ab + ac \).</li>
</ul>
<p><strong>Example:</strong></p>
<p>Simplify \( 3x + 4x - 5 \).</p>
<p>\[ 3x + 4x - 5 = 7x - 5 \]</p>
<p><strong>Practice:</strong></p>
<ul>
<li>Simplify \( 2x^2 + 3x - x^2 \).</li>
<li>Expand and simplify \( 2(x + 3) + 4(x - 1) \).</li>
</ul>
<h3>Solve Equations</h3>
<p><strong>Key Points:</strong></p>
<ul>
<li>Isolate the variable.</li>
<li>Perform the same operation on both sides.</li>
</ul>
<p><strong>Example:</strong></p>
<p>Solve \( 2x + 3 = 7 \).</p>
<p>\[ 2x + 3 = 7 \Rightarrow 2x = 4 \Rightarrow x = 2 \]</p>
<p><strong>Practice:</strong></p>
<ul>
<li>Solve \( 3x - 4 = 11 \).</li>
<li>Solve \( 5(x - 2) = 3x + 6 \).</li>
</ul>
<h3>Determine Equation of a Line (Given Information Like Graph, Parallel/Perpendicular to Another Line, etc.)</h3>
<p><strong>Key Points:</strong></p>
<ul>
<li>Use point-slope form \( y - y_1 = m(x - x_1) \).</li>
<li>Parallel lines have equal slopes; perpendicular lines have negative reciprocal slopes.</li>
</ul>
<p><strong>Example:</strong></p>
<p>Find the equation of a line parallel to \( y = 2x + 3 \) passing through (1, 2).</p>
<p><strong>Practice:</strong></p>
<ul>
<li>Find the equation of a line perpendicular to \( y = \frac{1}{2}x + 4 \) passing through (0, 0).</li>
<li>Find the equation of a line passing through (2, 3) and (4, 7).</li>
</ul>
</article>
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